临界预湿时二维Ising界面的局部和全局几何
Local and global geometry of the 2D Ising interface in critical pre-wetting
摘要：
考虑低温和正外场$λ$下的伊辛模型，$λ$位于$N\times N$盒上，Dobrushin边界条件在北、东、西边界上为正，在南边界上为负。如果为$λ= 0$，则分离正相和负相的界面是扩散的，具有$O(\sqrt N)$高度波动，并且模型完全润湿。在一阶场下，界面波动为$O(1)$，界面仅部分润湿，被钉扎到其南部边界。我们研究了$λ_N \downarrow 0$的临界预湿状态，其中高度波动预计将按$λ^{ -1/3}$缩放，并且预计重新缩放的界面将收敛于Ferrari-Spohn扩散。Velenik（2004）确定了界面下面积的数量级，直至对数校正。从那时起，更精细的功能，这样的接口已被确定在更简单的模型下的区域倾斜的随机游动。在本文中，我们解决了Velenik关于伊辛界面在临界预湿制度的细化功能的几个示意图。我们的主要结果是一个尖锐的约束上的一点高度波动，证明$e^{ - Θ(x^{3/2})}$上尾让人想起Tracy-Widom分布，捕获局部布朗振荡和全球的场效应之间的权衡。我们进一步证明了一个浓度估计的点的数量以上的接口达到一个大的高度。这些是用来推导各种几何性质的接口，包括该地区的顺序和尾巴，它的限制，和多对数前因子控制其最大高度波动。我们的论点结合了经典的输入从随机线表示的伊辛接口，与新的本地resches和耦合方案。
英文摘要：
Consider the Ising model at low-temperatures and positive external field $λ$ on an $N\times N$ box with Dobrushin boundary conditions that are plus on the north, east, and west boundaries and minus on the south boundary. If $λ= 0$, the interface separating the plus and minus phases is diffusive, having $O(\sqrt N)$ height fluctuations, and the model is fully wetted. Under an order one field, the interface fluctuations are $O(1)$ and the interface is only partially wetted, being pinned to its southern boundary. We study the critical pre-wetting regime of $λ_N \downarrow 0$, where the height fluctuations are expected to scale as $λ^{ -1/3}$ and the rescaled interface is predicted to converge to the Ferrari--Spohn diffusion. Velenik (2004) identified the order of the area under the interface up to logarithmic corrections. Since then, more refined features of such interfaces have only been identified in simpler models of random walks under area tilts. In this paper, we resolve several conjectures of Velenik regarding the refined features of the Ising interface in the critical pre-wetting regime. Our main result is a sharp bound on the one-point height fluctuation, proving $e^{ - Θ(x^{3/2})}$ upper tails reminiscent of the Tracy--Widom distribution, capturing a tradeoff between the locally Brownian oscillations and the global field effect. We further prove a concentration estimate for the number of points above which the interface attains a large height. These are used to deduce various geometric properties of the interface, including the order and tails of the area it confines, and the poly-logarithmic pre-factor governing its maximum height fluctuation. Our arguments combine classical inputs from the random-line representation of the Ising interface, with novel local resampling and coupling schemes.
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